Relaxed and logarithmic modules of $$\widehat{{{\mathfrak {s}}}{{\mathfrak {l}}}_3}$$
Bibliographic record
Abstract
Abstract In Adamović (Commun Math Phys 366:1025–1067, 2019), the affine vertex algebra $$L_k({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is realized as a subalgebra of the vertex algebra $$Vir_c \otimes \Pi (0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>⊗</mml:mo> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $$Vir_c $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> is a simple Virasoro vertex algebra and $$\Pi (0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Π</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a half-lattice vertex algebra. Moreover, all $$L_k({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -modules (including, modules in the category $$KL_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> , relaxed highest weight modules and logarithmic modules) are realized as $$Vir_c \otimes \Pi (0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>⊗</mml:mo> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case $${\mathfrak {g}}= {{\mathfrak {s}}}{{\mathfrak {l}}}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>s</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math> and present realization of the VOA $$L_k({\mathfrak {g}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for $$k \notin {\mathbb {Z}}_{\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∉</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> as a vertex subalgebra of $${\mathcal {W}}_ k \otimes {\mathcal {S}} \otimes \Pi (0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>⊗</mml:mo> <mml:mi>S</mml:mi> <mml:mo>⊗</mml:mo> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $${\mathcal {W}}_ k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> is a simple Bershadsky–Polyakov vertex algebra and $${\mathcal {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> is the $$\beta \gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmic modules. We prove the irreducibility of all our relaxed highest weight modules having finite-dimensional weight spaces (whose top components are Gelfand–Tsetlin modules). The irreducibility of relaxed highest weight modules with infinite-dimensional weight spaces is proved up to a conjecture on the irreducibility of certain $${\mathfrak {g}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -modules which are not Gelfand–Tsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of $${\mathcal {W}}_ k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> from Adamović et al. (Lett Math Phys 111(2), Paper No. 38, <jats:ext-li
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.024 | 0.004 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".